Fig. 2-9 Cross section of a doughnut shape plasma

Fig. 2-10 Time evolution of perturbation energy of ballooning modes with and without plasma rotation (alpha's are chosen such that the mode is stable and unstable in each figure) Plasma rotation shear damps the perturbation energy. This compensates for the growth, and the ballooning instability is suppressed.

Fig. 2-11 Schematic explanation for the effect of the plasma cross section (behavior at the growing phases is exaggerated) In the growing phases, D-shaping reduces both (1) the increasing rate of the perturbation energy and (2) the duration. The plasma rotation shear and the D-shaping combine to provide cooperative stabilization.

Confinement of high-pressure plasma is crucial for a fusion reactor. Magnetically confined plasma with a doughnut shape (torus) can swell the magnetic field at the outboard side (bad curvature region) (Fig. 2-9). If the pressure gradient exceeds a threshold, the swelling grows (ballooning instability). This instability can degrade the confinement performance. Thus, it has been extensively studied. A theory for the ballooning instability in a mechanical equilibrium without a plasma rotation has been established. Several studies also have been done for a rotating plasma. The rotation stabilizes the ballooning instability and increases the threshold for the pressure gradient. However, the mechanism of this stabilization has not been clarified. We have solved equations for the perturbation from the mechanical equilibrium as a part of the Numerical Experiment of Tokamak (NEXT) project, and have clarified the following. The perturbation energy is never damped in the absence of plasma rotation (Fig. 2-10(a)). Conversely, it damps periodically with time when plasma rotation shear exists (Fig. 2-10(b)). This damping compensates for the perturbation growth related to the bad curvature, and therefore the ballooning mode can be stabilized. D-shaping reduces both (1) the increasing rate of the perturbation energy and (2) the duration of the growing phases (Fig. 2-11) because of the relative reduction of the bad curvature region. D-shaping and the plasma rotation shear combine to provide cooperative (synergetic) stabilization. D-shaping is experimentally considered a key for confinement of high-pressure plasma. We have theoretically clarified that D-shaping combines with plasma rotation to provide stabilization.

Reference M. Furukawa et al., Geometrical Improvements of Rotational Stabilization of High-n Ballooning Modes in Tokamaks, Nucl. Fusion, 43, 425 (2003).

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